Mathematics - list 4
1. Compute indenite integrals
√
2
√
3
−x x
)dx, (4) xex dx,
(1) (x3 − 3x + 7)dx, (2) (2 3 x2 − x1 + x22 )dx, (3) ( 3x √
R 2
R x
R
R xα
(5) (x + 1) sin xdx
x ln xdx, where
α∈R
R , (6) e cosR xdx, (7) ln xdx, (8)
R
R
1
is a constant, (9) ln2 xdx, (10) sin 2x cos xdx, (11) sin2 xdx, (12) (2x+1)
2 dx,
√
R √
R
R
R√
2
5
2
3
(13)
3x + 1dx, (14) x x − 1dx, (15) x(2x +1) dx, (16) (3x +2) x + 2x + 1dx,
R
R
R
R x2
2
dx, (21)
(17) x22x+3
arctan xdx (19) xe−2x +1 dx, (20) √
5 3
+3x−1 dx, (18)
x +1
R
R
R
R
√
R −1
R
R
x
tan xdx, (23) sin√x x dx, (24) ex2x dx, (25) arctan
1+x2 dx, (26)
R
R x√
R
R √
3 −x
e +1
e ex + 1, (28) (2x + 1) sin 3xdx, (29)
dx, (30)
R ex 2
R
R
R ln ln x
2
x(x +1)ex dx,
(3x−1)e
dx, (31) (2x+3) ln(x+1)dx, (32)
x dxR, (33)
R √
R
(34) 2x x2 + 1 ln(x2 +1)dx, (35) sin(2x)esin x dx, (36) ln(sin2 x)(tan x)−1 dx.
R
√ cos x dx, (22)
1+sin x
√
3
ln x+1
dx, (27)
x
−3x
R
2. (∗ optional) Compute indenite integrals
x+1
+x−1
x+1
x−1
(1) x−1
dx, (2) x x+1
dx, (3) x2x+1
, (4) x2 +4x+5
dx, (5) (x+1)
2 (x+2) dx,
+x−2 dx
√
√
R
R
R
R
R
x
1+√x
x+1
x
dx
(6) (x2 +4)(x−1) dx, (7)
dx, (9) ex +e−x dx, (10) 2eex +1 dx,
x dx, (8)
1− x
R√
(11)
ex + 1dx.
R
R
2
R
R
R
3. (E1 ) Check which of the numbers I1 or I2 is greater:
R1
R1
2
(a) I1 = 0 8xe−4x dx and I2 = 0 xe−x dx,
(b) I1 =
Re
1
3 ln2 x
x dx
and I2 =
Re
1
x2 ln xdx.
√
4. (E) Let f (x) = | x − 1| for x ≥ 0.
(a) Determine local extremes of f .
(b) Find the area of the region bounded by the graph of f and the lines x = 0,
x = 4, y = 0.
5. (E) Let f (x) = | x1 − 1| for x > 0.
(a) Determine local extremes of f .
(b) Find the area of the region bounded by the graph of f and the lines x = e−1 ,
x = e, y = 0.
6. Sketch the region enclosed by the given curves and nd its area
√
(1) y = 3 − x, y = x + 1, y = 0, x = 4.
(2) y = x1 , y = x, y = 14 x, x > 0.
(3)
(4)
(5)
(6)
1 Means
x = y4 , y =
√
2 − x, y = 0.
y = x − x − 6 and y = −x2 + 5x + 14,
√
y = 2x2 − 1, y = x, y = −x.
2
y = xe−2x , y = 0, x =
1
2
that a task was on an exam.
1
7. (E) Let f (x) = x tet dt for x < 0.
(a) Find the formula for f (x) computing the denite integral.
(b) Determine the interval at which f increases slower (i.e., is increasing and concave).
R0
8. (E) Let f (x) = x3 (x4 − 16)3 .
R
(a) Calculate f (x)dx.
(b) Find the antiderivative of the function f (x) for which the minimum value on
the interval [1, 3] is equal to 5.
2